Computationally efficient algorithms may be applied for fast dictionary learning solving the convolutional sparse coding problem in the Fourier domain. More specifically, efficient convolutional sparse coding may be derived within an alternating direction method of multipliers (ADMM) framework that utilizes fast Fourier transforms (FFT) to solve the main linear system in the frequency domain. Such algorithms may enable a significant reduction in computational cost over conventional approaches by implementing a linear solver for the most critical and computationally expensive component of the conventional iterative algorithm. The theoretical computational cost of the algorithm may be reduced from O(M3N) to O(MN log N), where N is the dimensionality of the data and M is the number of elements in the dictionary. This significant improvement in efficiency may greatly increase the range of problems that can practically be addressed via convolutional sparse representations.
Legal claims defining the scope of protection. Each claim is shown in both the original legal language and a plain English translation.
1. A computer-implemented method, comprising: deriving efficient convolutional sparse coding in a frequency domain, by a computing system, within an alternating direction method of multipliers (ADMM) framework using fast Fourier transforms (FFTs); determining, by the computing system, coefficient maps of a signal or image vector s using the derived efficient convolutional sparse coding; and when stopping criteria are met, outputting the coefficient maps, by the computing system, as a sparse representation of s.
A computer-implemented method for efficient image or signal processing uses convolutional sparse coding in the frequency domain. It leverages an Alternating Direction Method of Multipliers (ADMM) framework, accelerated by Fast Fourier Transforms (FFTs), to decompose a signal or image into a sparse representation. The system calculates coefficient maps (representing the sparse components) of the input signal. Once a defined stopping criteria is met (indicating sufficient convergence), the system outputs these coefficient maps as the sparse representation of the original signal or image. This provides a computationally efficient way to represent the data using a small set of learned basis functions.
2. The computer-implemented method of claim 1 , wherein the coefficient maps are determined with an efficiency of O(MN log N), where N is a dimensionality of the data and M is a number of elements in a dictionary.
The efficient convolutional sparse coding method, which determines coefficient maps of a signal or image vector s using derived efficient convolutional sparse coding in a frequency domain, by a computing system, within an alternating direction method of multipliers (ADMM) framework using fast Fourier transforms (FFTs) and outputs the coefficient maps as a sparse representation of s when stopping criteria are met, achieves a computational efficiency of O(MN log N). Here, N represents the dimensionality of the input data (e.g., the number of pixels in an image), and M represents the number of elements (basis functions) in the dictionary used for sparse coding. This logarithmic scaling with data dimensionality makes the method suitable for large datasets.
3. The computer-implemented method of claim 1 , wherein the coefficient maps are computed using only inner products, element-wise addition, and scalar multiplication as vector operations.
The efficient convolutional sparse coding method, which determines coefficient maps of a signal or image vector s using derived efficient convolutional sparse coding in a frequency domain, by a computing system, within an alternating direction method of multipliers (ADMM) framework using fast Fourier transforms (FFTs) and outputs the coefficient maps as a sparse representation of s when stopping criteria are met, relies exclusively on basic vector operations: inner products (dot products), element-wise addition of vectors, and scalar multiplication. This restriction to simple operations makes the algorithm highly amenable to parallelization and optimization on various hardware platforms, leading to increased speed and efficiency.
4. The computer-implemented method of claim 1 , further comprising: precomputing, by the computing system, FFTs of a dictionary D and the signal or image vector s.
To further enhance the performance of the efficient convolutional sparse coding method, which determines coefficient maps of a signal or image vector s using derived efficient convolutional sparse coding in a frequency domain, by a computing system, within an alternating direction method of multipliers (ADMM) framework using fast Fourier transforms (FFTs) and outputs the coefficient maps as a sparse representation of s when stopping criteria are met, the system precomputes the FFTs of both the dictionary (D, containing the basis functions) and the input signal or image vector (s). This precomputation avoids redundant FFT calculations within the iterative ADMM loop, leading to significant speedups.
5. The computer-implemented method of claim 1 , further comprising: initializing auxiliary variables, by the computing system, to zero.
Before the iterative process begins in the efficient convolutional sparse coding method, which determines coefficient maps of a signal or image vector s using derived efficient convolutional sparse coding in a frequency domain, by a computing system, within an alternating direction method of multipliers (ADMM) framework using fast Fourier transforms (FFTs) and outputs the coefficient maps as a sparse representation of s when stopping criteria are met, the auxiliary variables used within the ADMM framework are initialized to zero. This initialization provides a starting point for the iterative optimization process, influencing the convergence speed and the final sparse representation.
6. The computer-implemented method of claim 1 , wherein while the stopping criteria have not been met, the method further comprises: computing, by the computing system, FFTs of auxiliary variables, frequency domain coefficient maps, inverse FFTs of the coefficient maps, and calculating the auxiliary variables; and updating auxiliary parameter ρ when convergence to a desired accuracy has not occurred.
Within the iterative loop of the efficient convolutional sparse coding method, which determines coefficient maps of a signal or image vector s using derived efficient convolutional sparse coding in a frequency domain, by a computing system, within an alternating direction method of multipliers (ADMM) framework using fast Fourier transforms (FFTs) and outputs the coefficient maps as a sparse representation of s when stopping criteria are met, the following steps are repeated until convergence: compute FFTs of auxiliary variables and frequency domain coefficient maps; compute inverse FFTs of the coefficient maps; calculate updated auxiliary variables based on these transforms. If the convergence to the desired accuracy isn't achieved, an auxiliary parameter ρ is also updated to influence the convergence behavior.
7. The computer-implemented method of claim 1 , wherein the computing system determines a set of coefficient maps in the frequency domain by v n = ρ - 1 ( b n - a n H b n ρ + a n H a n a n ) .
The efficient convolutional sparse coding method, which determines coefficient maps of a signal or image vector s using derived efficient convolutional sparse coding in a frequency domain, by a computing system, within an alternating direction method of multipliers (ADMM) framework using fast Fourier transforms (FFTs) and outputs the coefficient maps as a sparse representation of s when stopping criteria are met, calculates the coefficient maps (vn) in the frequency domain using the formula: vn = (1/ρ) * (bn - (a_nH * bn) / (ρ + a_nH * an)) * an, where a and b are intermediate variables, ρ is an auxiliary parameter influencing convergence, and a_nH denotes the Hermitian transpose of an. This equation represents a core step in the ADMM optimization within the frequency domain, solving for the sparse coefficients efficiently.
8. The computer-implemented method of claim 1 , further comprising: learning a dictionary D from a set of training data, wherein a FFT of D yields a dictionary in the frequency domain {circumflex over (D)} such that D ^ = ( d ^ 0 , 0 0 0 ⋯ d ^ 1 , 0 0 0 ⋯ 0 d ^ 0 , 1 0 ⋯ 0 d ^ 1 , 1 0 ⋯ 0 0 d ^ 0 , 2 ⋯ 0 0 d ^ 1 , 2 ⋯ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋱ ) where {circumflex over (D)} is concatenated as a set of block matrices and each block matrix is a diagonal.
The efficient convolutional sparse coding method, which determines coefficient maps of a signal or image vector s using derived efficient convolutional sparse coding in a frequency domain, by a computing system, within an alternating direction method of multipliers (ADMM) framework using fast Fourier transforms (FFTs) and outputs the coefficient maps as a sparse representation of s when stopping criteria are met, can further include a dictionary learning step. The dictionary (D) is learned from a set of training data. The FFT of this learned dictionary D results in a frequency domain representation (D-hat), structured as concatenated block matrices where each block matrix is diagonal. D-hat = ( d^0,0 0 0 ... d^1,0 0 0 ... 0 d^0,1 0 ... 0 d^1,1 0 ... 0 0 d^0,2 ... 0 0 d^1,2 ... ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋱ ). This learned dictionary provides basis functions optimized for the specific type of signal or image being processed.
9. The computer-implemented method of claim 8 , wherein D is a multi-scale dictionary.
The dictionary learning aspect of the efficient convolutional sparse coding method, where a dictionary D is learned from a set of training data and whose FFT yields a dictionary in the frequency domain {circumflex over (D)} such that D ^ = ( d ^ 0, 0 0 0 ⋯ d ^ 1, 0 0 0 ⋯ 0 d ^ 0, 1 0 ⋯ 0 d ^ 1, 1 0 ⋯ 0 0 d ^ 0, 2 ⋯ 0 0 d ^ 1, 2 ⋯ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋱ ) where {circumflex over (D)} is concatenated as a set of block matrices and each block matrix is a diagonal, utilizes a multi-scale dictionary. This means the dictionary contains basis functions at various scales or resolutions, enabling the representation of signal or image features at different levels of detail. A multi-scale dictionary improves the ability of the sparse coding method to capture both fine-grained and coarse-grained structures in the input data.
10. The computer-implemented method of claim 1 , further comprising: computing, by the computing system, a dictionary in a frequency domain ĝ m ∀m using ŷ k,m as coefficient maps and using an iterated Sherman-Morrison algorithm for a dictionary update; and outputting, by the computing system, a dictionary {g m } when stopping tolerances are met.
The efficient convolutional sparse coding method, which determines coefficient maps of a signal or image vector s using derived efficient convolutional sparse coding in a frequency domain, by a computing system, within an alternating direction method of multipliers (ADMM) framework using fast Fourier transforms (FFTs) and outputs the coefficient maps as a sparse representation of s when stopping criteria are met, includes dictionary learning. The system computes a dictionary in the frequency domain (g-hat_m for all m) using the coefficient maps (y_k,m) and applies an iterated Sherman-Morrison algorithm for updating the dictionary. When the stopping tolerances for dictionary learning are met, the learned dictionary {g_m} is output. The Sherman-Morrison algorithm provides an efficient way to update the inverse of a matrix after a rank-one update, which is useful for online dictionary learning.
11. The computer-implemented method of claim 10 , further comprising: interleaving, by the computing system, updates on sparse coding and dictionary learning such that g m represents the dictionary in sparse coding steps and y k,m represent sparse coding in dictionary steps; and outputting, by the computing system, coefficient maps {y m } when the stopping tolerances are met.
The efficient convolutional sparse coding method, which computes a dictionary in a frequency domain ĝ m ∀m using ŷ k,m as coefficient maps and using an iterated Sherman-Morrison algorithm for a dictionary update and outputs a dictionary {g m } when stopping tolerances are met, alternates updates on sparse coding (determining coefficient maps) and dictionary learning. This interleaving means that `g_m` represents the dictionary during sparse coding steps, and `y_k,m` represents the sparse coefficients during dictionary learning steps. The system iteratively refines both the sparse coding and the dictionary until the stopping tolerances for both are met, at which point the final coefficient maps {y_m} are output. This interleaved approach allows for mutual refinement, resulting in a more accurate and efficient sparse representation.
Cooperative Patent Classification codes for this invention. Click any code to explore related patents in that topic.
March 25, 2015
June 20, 2017
Browse 5M+ US patents with plain-English claim translations and AI-generated analysis.